About Infinity

I need to know about how mathematics works with infinity. I am currently reading up on this but it could take me a few weeks to finish this book.

Ex: You are created, standing in space. You are flipping a coin over and over for infinity.

You thus flip the coin an infinite amount of times.
Heads is flipped an infinite amount of times,
Tails is flipped an infinite amount of times.

Now figure out the probablility of flipping heads... 100%??? Infinity / Infinity. I need to know if there is some way to use 2(infinity) or something like this in a formula, it makes sence. Is there some way to distinguish between 2 infinite amounts?

Another example would be the paradox with the infinite whole numbers, each being the perfect square root of another number.

One way to handle it here is to pick some finite subsets of the state space and look at the probability of the event there. If these sets are increasing and eventually cover the total space, and the limit of the probablities for the finite subsets exists and is independent of the way you chose the covering then you might think of that being probability.

As it is you've not defined the question sufficiently well. Remember that in probability you need a state space , some collection of subsets of the state space (allowable events), and a measure from the set of subsets satisfying certian rules (the probability of an event occurring).

In the cases you give you have failed to give all that information and as such it doesn't form the basis for a proper question about probability. There are many ways to assign measures, deciding if its realistic is up to you, but seeing as you can't toss a coin an infinite number of times it doesn't really matter here. And by infinite you mean a countable number of times?

The naive way you're treating probability isn't how it works, so any problems are possibly not with your notion of infinity but with your knowledge of probability.

Ex: You are created, standing in space. You are flipping a coin over and over for infinity.

You thus flip the coin an infinite amount of times.
Heads is flipped an infinite amount of times,
Tails is flipped an infinite amount of times.

Now figure out the probablility of flipping heads... 100%??? Infinity / Infinity. I need to know if there is some way to use 2(infinity) or something like this in a formula, it makes sence. Is there some way to distinguish between 2 infinite amounts?

Another example would be the paradox with the infinite whole numbers, each being the perfect square root of another number.

1 2 3 4 5 6 ... infinity
1 4 9 16 25 36 ... infinity

Infinity is pure evil and can "override" many of the laws of sience.
For one:
A line is made out of an infinite amount of point, though, a point has only position but no extension in space, thus, the length of the line each point contributes with is 0, still, the line has length.
l = 0 * infinity.
(so much for the anything multiplied with 0 is 0)

To begin with, a cource in derivates and intergrals may be good since you get to work with it somewhat there.

Also, the heads/times flipped won't give you a meaningful answer since,
infinity/infinity can mean any value, it's undefined (just like 0/0 is).
2 * infinity doesn't work as such either, no matter what you add to infinity, it's still infinite.

In the case of the coin flipping, there are a few ways to calculate it.
For starters, you can flip it once, then again, and again, noting each value, then, as the number of flips grows
heads/flips
goes towards a certain value.
If you do this empericly, it's impossible to assign a 100% sure value (though if you follow actual logical rules and rely on the probablity rather on the results of probability you may reach the goal value). In the case where the flipping weights more on formulas and alike, you're in many cases able to calculate what the value would be at infinite, not by actually counting it for infinite by something like go infinity close to infinity.

*cough* since infinity isn't a number that one may just abuse by multiplying like that and since length isn't just number of points in the set times the length of the point, you issues aren't that mathematics has problems, but more that you have issues with mathematics. it may go against \emph{your} intuition but that isn't important.

If infinity is a number with no limit, then why can infinity/infinity not be defined? It should equal 1, obviously. You cant say that 1 infinity may be greater or less than the other one because infinity is THE GREATEST. I personally think that we should be able to use things such as "2infinity" and infinity/2 in mathematics. The coin flipping example shows this, the coin is obviously flipped 2 times as much as each heads and tails is flipped.